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4PM1/02 Jun 2022, Question 3: Unit Vectors and i, j Components

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 9 marks

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The question, in short

Vectors OA = πi − 4j and OB = i + (2p + 1)j with √2|OA| = |OB| and p > 0; find p, then a unit vector parallel to AB.

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Part (a) · 4 marks

Use the given magnitude relation √2|OA| = |OB| with p > 0 to solve for p.

  1. 1|OA| = √(p² + 16), |OB| = √(1 + (2p + 1)²) = √(4p² + 4p + 2)
  2. 22(p² + 16) = 4p² + 4p + 2
  3. 32p² + 4p − 30 = 0 ⇒ p² + 2p − 15 = 0
  4. 4(p + 5)(p − 3) = 0, p > 0 so p = 3

Answerp = 3

Part (b) · 5 marks

With the found p, form vector AB and give a unit vector in its direction.

  1. 1OA = 3i − 4j, OB = i + 7j
  2. 2AB = OB − OA = −2i + 11j
  3. 3|AB| = √(4 + 121) = √125 = 5√5
  4. 4Unit vector = (1/(5√5))(−2i + 11j)

Answer(1/(5√5))(−2i + 11j) (or its negative)

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